16.3. In times before global positioning systems (GPS) or navigation devices, hikers would. The Inverse Undoes What a Function Does

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1 The Inverse Undoes What a Function Does Inverses of Linear Functions.3 Learning Goals In this lesson, you will: Determine the inverse of a given situation using words. Determine the inverse of a function numerically using a table. Determine the inverse of a function using algebra. Determine the inverse of a function using graphical representations. Calculate compositions of functions. Use compositions of functions to determine whether functions are inverses. Key Terms inverse operation inverse function composition of functions In times before global positioning systems (GPS) or navigation devices, hikers would routinely leave markers (sometimes called bread crumbs) on the ground to mark where they walked. This was helpful especially if hikers were traveling through thick forests. Nowadays, this type of tracking is primitive, but the terminology still remains and this terminology goes beyond hiking. In fact, bread crumb terminology is commonly used in creating and navigating through web sites. The principle is simple: just ensure that the user retraces their steps to get back to a home page. Can you think of other activities in which bread crumbing is a common practice? 1151

2 Problem 1 The Reverse Is the Inverse Recall that a relation is the mapping between a set of inputs and a set of outputs, and a function is a special type of relation in which every element of its domain is associated with exactly one element of its range. 1. Every member on a high school football team is assigned a jersey number. a. Let the members of the football team represent the domain, and let the assigned numbers represent the range. Is this relation a function? Explain why or why not. b. Consider reversing the problem situation so that the assigned numbers represent the domain, and the members of the football team represent the range. Is this reverse relationship a function? Explain why or why not. 2. Each student in your school chooses his or her favorite color. a. Let the students in your school represent the domain, and let all of the colors represent the range. Is this relation a function? Explain why or why not. b. Reverse the problem situation. What is the domain of the reverse relationship? What is the range? c. Is the reverse relationship a function? Explain why or why not Chapter Other Functions and Inverses

3 You determined the reverse of general relationships in Questions 1 and 2. When determining the reverse for more specific situations, think about how to undo the situation. Undoing, working backwards, or retracing steps to return to an original value or position is referred to as using the inverse operation. For instance, subtracting 4 from a number is the inverse of adding 4 to a number, because the subtraction of 4 undoes the addition of 4 to a number. 3. Write a phrase, expression, or sentence to describe the inverse of each given situation. a. Open a door. b. Turn off a light. c. Get into the deep end of a pool and swim to the shallow end of the pool. d. Walk 2 blocks east and then 3 blocks south.?4. Consider the given situation. Multiply a number by 3 and then subtract 5. Marilyn says that the inverse is divide by 3 and then add 5. Jake says that the inverse is add 5 and then divide by 3. Who is correct? Explain how you determined which student is correct..3 Inverses of Linear Functions 1153

4 Problem 2 Converting Dollars to Lira Miguel is planning a trip to Turkey. Before he leaves, he wants to exchange his money to the Turkish lira, the official currency of Turkey. The exchange rate at the time of his trip is 2 lira per 1 U.S. dollar. 1. What are the independent and dependent quantities? 2. Complete the table of values to show the currency conversion for U.S. dollars to Turkish lira. U.S. Currency (dollars) Turkish Currency (lira) Write an equation to represent the number of lira in terms of the number of U.S. dollars. Let r represent the number of lira, and let d represent the number of U.S. dollars. 4. What is the inverse of this problem situation? 5. What are the independent and dependent quantities of the inverse of the problem situation? How do these quantities compare to the quantities in Question 1? 6. Complete the table of values to show the inverse of the problem situation. Turkish Currency (lira) Chapter Other Functions and Inverses

5 7. Compare the tables in Questions 2 and 6. What do you notice? 8. Write an equation for the inverse of the problem situation. Use the same variable representations, where r represents the number of lira, and d represents the number of U.S. dollars. Does this equation represent a function? 9. Explain how you determined your equation in Question 8. The equation you determined in Question 8 is called an inverse function. Recall that a function takes an input value, performs some operation(s) on this value, and creates an output value. The inverse function takes the output value, performs some operation(s) on this value, and arrives back at the original function s input value. In other words, an inverse function is a function that undoes another function. 10. Substitute your equation in Question 8 into an equivalent expression in your equation in Question 3, then simplify. What do you notice? The notation for inverse, f 1 (x), 2 does not mean the same thing as x 1. The expression x can be rewritten as x 1_ ; however, f 1 (x) 2 cannot be rewritten, because it is only used as notation. In other words, f 1 (x) 2 fi 1 f (x). For a function f(x), the notation for its inverse is f 21 (x). 11. Use function notation to represent the number of lira f(x) in terms of the number of U.S. dollars, x. Given a function f(x), you can determine the inverse function f 21 (x) algebraically by following the steps shown. Step 1: Replace the function f(x) with another variable, generally y. Step 2: Switch the x and y variables in the equation. Step 3: Solve for y. Step 4: If y is a function, replace y with f 21 (x)..3 Inverses of Linear Functions 1155

6 12. Determine the inverse of the function you wrote in Question 11 using the steps in the worked example to verify the equation you wrote in Question 8 was the inverse of the problem situation. Problem 3 Graphs of Inverses of Functions and Paper Folding Consider the function f(x) from Problem 2 and its inverse f 21 (x). 1. Complete the steps shown. Step 1: Graph f(x), f 21 (x), and the line y 5 x together on the grid shown. Step 2: Heavily trace the graph of f(x) with a pencil. Step 3: Fold the graph along the line y 5 x, and rub the paper so that the image of the graph of f(x) appears. Put down that pen a pencil is the best tool for this job! y x When they say heavily trace, they mean heavily trace! You may have to go back and trace it a few times before the image appears Chapter Other Functions and Inverses

7 2. Describe the type of transformation you performed. 3. How is the transformation that you performed in this lesson different from transformations you performed in previous lessons? Do you remember the different types of transformations of geometric figures? 4. Compare the image you created and the graph of f 21 (x). a. What do you notice about the image and the graph of f 21 (x)? b. What does this tell you about the line y 5 x? c. What does this tell you about the graph of the inverse of a function? 5. For each function and a given point on the graph of the function, determine the corresponding point on the graph of the inverse of the function. a. Given that (3, 2) is a point on the graph of g(x), what is the corresponding point on the graph of g 21 (x)? b. Given that (21, 0) is a point on the graph of h(x), what is the corresponding point on the graph of h 21 (x)? c. Given that (a, b) is a point on the graph of f(x), what is the corresponding point on the graph of f 21 (x)?.3 Inverses of Linear Functions 1157

8 Problem 4 Compositions of Functions 1. Damien coaches a neighborhood softball team. His team won their division, so he wants to order division champion ball caps for his players. The cost of each cap is $8. There is an additional 7% sales tax for each cap. a. Write a function c(x) for the cost of x caps before sales tax. b. Write a function s(x) for the cost with sales tax if the caps cost x dollars. c. Write a function t(x) for the total cost of x caps with sales tax. The function you wrote in Question 1 part (c) is a composition of the functions in parts (a) and (b). A composition of functions is the combination of functions such that the output from one function becomes the input for the next function. So, one function is substituted as the input value of the other function. Keep in mind that f(g(x)) does not imply the operation of multiplication. It is simply the notation used to describe the composition of f (x) with g(x). The composition of function f(x) composed with g(x) is denoted (f 8 g)(x) or f(g(x)). It is read as f composed with g(x) or f of g(x). Suppose that f(x) 5 3x and g(x) 5 x 1 1. You can determine the composition (f 8 g)(x), or f(g(x)), using substitution. Step 1: Substitute (x 1 1) for g(x). f(g(x)) 5 f(x 1 1) Step 2: Substitute (x 1 1) into f(x) 5 3x in place of x. 5 3(x 1 1) Step 3: Simplify. 5 3x 1 3 You can also determine the composition (g 8 f )(x), or g(f(x)). Step 1: Substitute (3x) for f(x). g(f(x)) 5 g(3x) Step 2: Substitute (3x) into g(x) 5 x 1 1 in place of x. 5 3x Chapter Other Functions and Inverses

9 2. For the functions you wrote in Question 1, verify that the composition c(s(x)) is equal to t(x). You can use compositions of functions to determine whether two functions are inverses. If f(x) and g(x) are inverse functions, then f(g(x)) 5 g(f(x)) 5 x. 3. Are the functions f(x) 5 3x and g(x) 5 x 1 1 from the worked example inverse functions? Justify your answer. 4. You wrote a function f(x) and determined its inverse f 21 (x) back in Problem 2. Use compositions to verify that f(x) and f 21 (x) are inverse functions.?5. Consider the function f(x) 5 3x 2 1. Rebecca says that the inverse function is f 21 (x) 5 1 x Damon says that the inverse function is f 21 (x) 5 x Who is correct? Explain your reasoning and show all your work..3 Inverses of Linear Functions 1159

10 Talk the Talk 1. Determine the inverse of each function. Is the inverse also a function? Explain why or why not. a. f(x) 5 0.6x 2 2 b. g(x) 5 8 x Determine whether the functions j(x) x and k(x) x 1 2 are inverses. If so, explain how you know. If not, determine each function s inverse. 10 Chapter Other Functions and Inverses

11 3. Sketch the inverse of the given function on the same graph as the function. Is the inverse also a function? Explain why or why not. a. y x b. y x Is the inverse of every linear function also a function? Explain why or why not..3 Inverses of Linear Functions 11

12 5. How can you determine if an inverse exists given a linear function? 6. Can a linear function and its inverse be the same function? If so, provide an example. If not, explain why not. 7. Complete the graphic organizer on the next page. Write the definition for the inverse of a linear function. Then describe how to determine the inverse of a function algebraically, graphically, and numerically. Be prepared to share your solutions and methods. 12 Chapter Other Functions and Inverses

13 Definition Algebraic Description Inverses of Linear Functions Graphical Description Numeric Description (Table of Values).3 Inverses of Linear Functions 13

14 14 Chapter Other Functions and Inverses